Definition:Local Trivialization
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Definition
Let $\struct {E, M, \pi, F}$ be a fiber bundle.
Let $\pr_1: M \times F \to M$ be the first projection on $M \times F$.
By definition of fiber bundle, for every point $m \in M$ there exists an open neighborhood $U$ of $m$ and a homeomorphism:
- $\chi: \pi^{-1} \sqbrk U \to U \times F$
such that:
- $\pi {\restriction}_U = \pr_1 \mathop \circ \chi$
where $\pi {\restriction}_U$ is the restriction of $\pi$ to $U$.
Then the ordered pair $\struct {U, \chi}$ is called a local trivialization of $E$ over $U$.
Also see
Sources
- 2003: John M. Lee: Introduction to Smooth Manifolds: $\S 10$: Fiber Bundles